Morse plot

Fig. 20. Bond scission activation energy and lifetime (Tt) plotted as a function of applied force. The solid curve is derived from Eq. (65) based on the Morse potential, the other data are redrawn from Ref. [101]. The upper abscissa gives the overall elastic strain before failure. The numbers indicate the minimum chain lengths which will fail at a particular force...

It is better than Morse function, because multiplying the exponential term, bypolynomial convergence can be checked by adding more terms by polynomial. This function can be linearised. A plot of In V versus r will give the value of D and o. ... 

The corresponding EDFs for both potentials calculated for the same test set of the Fe(II) complexes are plotted in Figs. 5, 8. The systematic errors in all cases are close to zero, as can be seen from the parameters of the EDFs for the EHCF/MM with the Morse potential — r = —0.005, (7=0.056 with the NR potential — /r =0.002, (7=0.052. However, although somewhat improved the stiffness of the Morse potential again manifests itself the systematic errors for the separate LS and HS sets do exist and only approximately cancel each other in the total set, whereas for the NR... 

The initial exploration in this unit requires the students to compare the trajectories calculated for several different energies for both Morse oscillator and harmonic oscillator approximations of a specific diatomic molecule. Each pair of students is given parameters for a different molecule. The students explore the influence of initial conditions and of the parameters of the potential on the vibrational motion. The differences are visualized in several ways. The velocity and position as a function of time are plotted in Figure 2 for an energy approximately 50% of the Morse Oscillator dissociation energy. The potential, kinetic and total energy as a function of time are plotted for the same parameters in Figure 3. 

Figure 9. Vibrational level spacing for HCP X S, A2G(D2) = G(0, u2 + 1,0)G(0, u2L0) plotted (open circles) vs. the bending quantum number vj. Also plotted l(open squares) is the reduced A2G, A2G(o2) - 2jc22i>2, which would be a horizontal line for a Morse oscillator.

Fig. 12.4. ln(r/n) plotted versus [m(en - en i)]x 2 for the vibrational predissociation of HeCl2. m is the reduced mass of the van der Waals molecule and en — en i is the energy spacing between adjacent levels of the Morse oscillator. Note that n increases from the right- to the left-hand side Adapted from Cline, Evard, Thommen, and Janda (1986). 

Figure 3.14. Stabilities of calcite, and synthetic (closed squares) and biogenic (closed circles) magnesian calcites as a function of composition. Stabilities are expressed as -log IAPMg-Calcite- The curve is a hand-drawn "best" fit to the synthetic data. Also plotted are the results of precipitation experiments by Mucci and Morse (1984, open squares) and biogenic dissolution experiments by Walter and Morse (1984a, open circles). (After Bischoff et al., 1987.)...

Figure 4.15. A plot of weight % calcium carbonate versus percent original calcium carbonate dissolved. For sediments of high initial calcium carbonate content, little change in weight % takes place until approximately 50 percent of the carbonate has dissolved. Small variations in initial calcium carbonate concentration can lead to large variations in calcium carbonate concentration at high dissolution values. (After Morse, 1973.)...

One of the most controversial areas of carbonate geochemistry has been the relation between calcium carbonate accumulation in deep sea sediments and the saturation state of the overlying water. The CCD, FL, R0, and ACD have been carefully mapped in many areas. However, with the exception of complete dissolution at the CCD and ACD, the extent of dissolution that has occurred in most sediments is difficult to determine. Consequently, it is generally not possible to make reasonably precise plots of percent dissolution versus depth. In addition, the analytical chemistry of the carbonate system (e.g., GEOSECS data) and constants used to calculate the saturation states of seawater have been a source of almost constant contention (see earlier discussions). Even our own calculations have resulted in differences for the saturation depth in the Atlantic of close to 1 km (e.g., Morse and Berner, 1979 this book). 

Samuel Morse, "The Present Attempt to Dissolve the American Union A British Aristocratic Plot" (New York John F. Trow, 1862) Samuel Morse, "A Foreign Conspiracy against the Liberties of the United States" (New York originally published by the New York Observer, 1835) see also the soon-to-be-published book, The First American Intelligence Service (New York Campaigner Publications). Morse signed all his published articles under the name "Brutus."... 

It has been emphasized that these calculations of activation energies are only approximate, commonly involving an error of 3,000 calories or more. Frequently the data for the constants D, r0 and co0 are not known with accuracy and the assumption of the fourteen per cent coulombic energy and the plotting of the empirical Morse curves are subject to considerable uncertainty. In some cases the calculations are quite tedious but frequently some type of symmetry greatly reduces the labor of computations and after some experience one finds it sufficient to select for computation only a limited number of points in the neighborhood of the... 

As mentioned above, HOSi(OA)3 may be taken as the simplest cluster model of the terminal hydroxyl group in silicas. Indeed, even with this cluster CNDO/BW provided a quite satisfactory description of the lower part of the curve representing potential energy as a function of the OH stretching vibration coordinate ROH (Fig. 2) (48,49). The respective experimental curve was plotted by Kazansky et al. (49) based on the analysis of the fundamental frequency vOH and the first overtone of the characteristic OH stretching vibration in terms of the Morse potential function. The frequencies of the second and third overtones were also determined in that work, and it was shown that the Morse potential reproduced well the potential curve within a rather wide range of ROH. 

Shin [24] has utilized a semiclassical approach to examine orientational effects on vibrational excitation. He treats the problems of (XX-A) and (XY-A) collisions, where A is an atom interacting with atoms of a diatomic molecule XX or XY through a Morse potential, and also the problems of (XX-XX) collisions and (XY-XY) collisions. The orientation dependence of the transition probabilities for Oa-Ar, 02-02, HBr-Ar, and HBr-HBr are plotted in polar diagrams. It is found, for example, that in the latter case only a very small range of angles gives the greatest contribution, namely, around that orientation in which the two H atoms lie between the Br atoms. 

In an earlier unit of this course, you were introduced to plots in which the energy possessed by a system of two atoms is shown as a function of the distance between them. These Morse curves, as they are sometimes called, are quite useful in defining certain properties of a chemical bond. 

Direct printout of Mathematica commands to calculate and plot wavefunctions and electronic transition intensity factors for a diatomic molecule (I2) using harmonic and Morse oscillator wavefunctions. See text for discussion. A Maple version of this calculation can be found on the Maple applications website. 

Use Mathcad or some other symbolic algebra program to solve the Ai secular determinant of Table 41-2 in a manner similar to that shown in Fig. 5 for the Bi determinant. Modify the Mathematica commands shown in Fig. 6 to see the square of the I2 harmonic oscillator and Morse wavefunctions and their overlap product for v" = 2 and v = 0, 5, 10, 15, 20, and 25. Obtain plots of these results and discuss the trends that you see. Repeat the exercise for v = 40 and v" = 0, 1, 2, 3, 4, and 5 and note the dramatic intensity variations for the Morse oscillator. Emission from this state, which can be populated by the 520.8-nm krypton ion laser line, is strong to even v" levels but is very weak to odd v" levels (up to about v" = 30). 

Calculate the harmonic force constant 4 and the Morse parameter j8 for the two states. Using the known r J value (0.2666 nm), plot the Morse curve for the ground electronic state of I2. Compare this with the harmonic potential calcnlated from k. ... 

Determine v (v ) from your spectrum and then r" - from this expression. Compare the resultant value of r with the literature value of 0.3025 nm obtained by analysis of the rotational structure of the electronic spectrum. Include the Morse potential curve for the upper state on your plot for the X state and comment on the differences in the various parameters determined for the two states. 

Knowing these parameters, one can plot the t/ vs. r curve for the stretch of a proton along one ofthep orbitals of oxygen. This plot is known as a.Morse curve and is shown in Fig. 4.124 for our system. The energy trajectory under study here pertains to the reaction shown in Fig. 4.125. Because of the symmetry of the system, one has an identical curve for the approach of the proton to the second water molecule. Thus,... 

Figure 11. Plot of the Peterson (41) and Berger (42) results for their water-column dissolution experiments in the Central Pacific Ocean, and the Morse and Berner (45) laboratory experiments as a function of equivalent depth. The depth of the lysocline was calculated from the data of Bramlette (49) (after Bef. 45).

Plots of the functions R R vs. for Ka =10 are shown in Fig.4. A point on the R curve corresponds to H2-He collisions on the Morse potential surface. One can see that the attractive enhancement of the VR cross section under consideration comes roughly evenly from resonance and potential effects. 

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